## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 5 (1978), 705-730.

### Sufficient Statistics and Extreme Points

#### Abstract

A convex set $M$ is called a simplex if there exists a subset $M_e$ of $M$ such that every $P \in M$ is the barycentre of one and only one probability measure $\mu$ concentrated on $M_e$. Elements of $M_e$ are called extreme points of $M$. To prove that a set of functions or measures is a simplex, usually the Choquet theorem on extreme points of convex sets in linear topological spaces is cited. We prove a simpler theorem which is more convenient for many applications. Instead of topological considerations, this theorem makes use of the concept of sufficient statistics.

#### Article information

**Source**

Ann. Probab., Volume 6, Number 5 (1978), 705-730.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995424

**Digital Object Identifier**

doi:10.1214/aop/1176995424

**Mathematical Reviews number (MathSciNet)**

MR518321

**Zentralblatt MATH identifier**

0403.62009

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J50: Boundary theory

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A25 28A65

**Keywords**

60-02 Extreme points sufficient statistics Gibbs states ergodic decomposition of an invariant measure symmetric measures entrance and exit laws excessive measures and functions

#### Citation

Dynkin, E. B. Sufficient Statistics and Extreme Points. Ann. Probab. 6 (1978), no. 5, 705--730. doi:10.1214/aop/1176995424. https://projecteuclid.org/euclid.aop/1176995424